(1.9) |

Let us narrow our focus to those functionals that preserve the linear dependencies of the kets upon which they operate. Not surprisingly, such functionals are termed

where and are any two kets in a given ket space.

Consider an
-dimensional ket space [i.e., a finite-dimensional, or
denumerably infinite dimensional (i.e.,
), space].
Let the
(where
runs from 1 to
)
represent
independent kets in this space.
A general ket can be written^{1.1}

where the are an arbitrary set of complex numbers. The functional can satisfy Equation (1.10) for all vectors in the ket space only if

where the are a set of complex numbers relating to the functional.

Let us define basis functionals that satisfy

Here, the

(1.14) |

But, this implies that the set of all possible linear functionals acting on an -dimensional ket space is itself an -dimensional vector space. This type of vector space is called a

(1.15) |

where DC stands for

There are an infinite number of ways of setting up the correspondence between vectors in a ket space and those in the related bra space. However, only one of these has any physical significance. (See Section 1.11.) For a general ket , specified by Equation (1.11), the corresponding bra is written

where the are the complex conjugates of the . is termed the

where is a complex number. More generally,

(1.18) |

Recall that a bra is a functional that acts on a general ket, and spits out a complex number. Consider the functional that is dual to the ket

acting on the ket . This operation is denoted . However, we can omit the round brackets without causing any ambiguity, so the operation can also be written . This expression can be further simplified to give . According to Equations (1.11), (1.13), (1.16), and (1.19),

Mathematicians term the

(See Exercise 3.) Consider the special case where . It follows from Equations (1.20) and (1.21) that is a real number, and that

The equality sign holds only if is the null ket [i.e., if all of the are zero in Equation (1.11)]. This property of bras and kets is essential for the probabilistic interpretation of quantum mechanics, as will become apparent in Section 1.11.

Two kets
and
are said to be *orthogonal*
if

(1.23) |

which also implies that .

Given a ket
, which is not the null ket,
we can define a *normalized ket*,
, where

(1.24) |

with the property

(1.25) |

Here, is known as the

It is possible to define a dual bra space for a ket space of non-denumerably infinite dimensions in much the same manner as that just described. The main differences are that summations over discrete labels become integrations over continuous labels, Kronecker delta symbols become Dirac delta functions, completeness must be assumed (it cannot be proved), and the normalization convention is somewhat different. (See Section 1.15.)